Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 167 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4.42·5-s + 1.88·7-s + 9-s + 1.84·11-s − 5.16·13-s − 4.42·15-s + 0.0891·17-s + 6.45·19-s + 1.88·21-s − 3.21·23-s + 14.6·25-s + 27-s + 3.02·29-s − 10.4·31-s + 1.84·33-s − 8.34·35-s − 1.51·37-s − 5.16·39-s − 7.82·41-s − 7.61·43-s − 4.42·45-s − 11.5·47-s − 3.45·49-s + 0.0891·51-s − 2.84·53-s − 8.17·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.98·5-s + 0.712·7-s + 0.333·9-s + 0.556·11-s − 1.43·13-s − 1.14·15-s + 0.0216·17-s + 1.48·19-s + 0.411·21-s − 0.669·23-s + 2.92·25-s + 0.192·27-s + 0.560·29-s − 1.87·31-s + 0.321·33-s − 1.41·35-s − 0.248·37-s − 0.827·39-s − 1.22·41-s − 1.16·43-s − 0.660·45-s − 1.68·47-s − 0.492·49-s + 0.0124·51-s − 0.390·53-s − 1.10·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 2004,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 + T \)
good5 \( 1 + 4.42T + 5T^{2} \)
7 \( 1 - 1.88T + 7T^{2} \)
11 \( 1 - 1.84T + 11T^{2} \)
13 \( 1 + 5.16T + 13T^{2} \)
17 \( 1 - 0.0891T + 17T^{2} \)
19 \( 1 - 6.45T + 19T^{2} \)
23 \( 1 + 3.21T + 23T^{2} \)
29 \( 1 - 3.02T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + 1.51T + 37T^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 + 7.61T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 2.84T + 53T^{2} \)
59 \( 1 + 5.92T + 59T^{2} \)
61 \( 1 - 4.29T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 0.174T + 73T^{2} \)
79 \( 1 - 2.45T + 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 + 18.3T + 89T^{2} \)
97 \( 1 + 3.13T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.490862562628386537458980171243, −8.007066634067260035822074839316, −7.33610297813236526247750640264, −6.87821667416014813987274828115, −5.17272213531317762740741372467, −4.63182857947169563066807306593, −3.67715217455151757843915691562, −3.07600684168264929405032420356, −1.59539316550349475434123811209, 0, 1.59539316550349475434123811209, 3.07600684168264929405032420356, 3.67715217455151757843915691562, 4.63182857947169563066807306593, 5.17272213531317762740741372467, 6.87821667416014813987274828115, 7.33610297813236526247750640264, 8.007066634067260035822074839316, 8.490862562628386537458980171243

Graph of the $Z$-function along the critical line